Can local duality for elliptic curves be proven with "big rings"?

Question

From Exercise 5.14, Ch. V of Silverman's "Advanced Topics in the Arithmetic of Elliptic Curves", I learned that the local duality for elliptic curves over $p$-adic fields can be proven for Tate curves by a relatively easy argument in Galois cohomology. Essentially, when the elliptic curve is $E_q = G_m / q^Z$ over a $p$-adic field $K$, one can find various long exact sequences connecting the Galois cohomology $H^1(K, E_q(\bar K))$ to the cohomology of $G_m$, $Q/Z$, etc., which are well-known by class field theory.

Without being an expert in $p$-adic cohomology, Hodge theory, etc., I know that by passing to a big ring ($B_{dR}$ will work), one can find a pair of periods for an elliptic curve over $K$ with good reduction. There might not be any interesting nontrivial uniformization of such elliptic curves, but the periods carry the information instead.

So can one exhibit (some of) the duality between $H^1(K, E(\bar K))$ and $Hom(E(K), Q/Z)$ when $E$ has good reduction, by using a big ring like $B_{dR}$?

When I see period rings, they are always used as linear algebraic gadgets. But since $B_{dR}$ is a $K$-algebra with Galois action, might someone consider $H^1(K, E(B_{dR}))$? In other words, rather than taking a linear algebraic gadget over $\bar K$, and tensoring up to $B_{dR}$, might one study a variety over $\bar K$ and base change to $B_{dR}$ or at least take $B_{dR}$-points? Might $H^1(K, E(B_{dR}))$ pick up the Weil-Chatelet group in the spirit of my first question?

Any references including $B_{dR}$-points of varieties would be greatly appreciated, as well as answers to the questions.

Answer 1

If I've understood correctly what you want, this is in the Bloch–Kato article in volume 1 of the Grothendieck Festschrift. The local duality result (I think) you are talking about is at the top of page 353 and the generalization is Proposition 3.8. Basically, you can show that if you have an abelian variety $A$ over $K$ with Tate module $T$, then the image of the Kummer map $$A(K)\rightarrow H^1(K,T)$$ is the Bloch–Kato Selmer group $H^1_f(K,T)$ (which is defined using $B_{\text{cris}}$; basically a derived functor of tensoring with $B_\text{cris}$ and taking Galois invariants). For a general $p$-adic Galois representation on a finite free $O_K$-module $T$, one can still define $H^1_f(K,T)$. One shows that under the usual local Tate duality $$H^1(K,T)\times H^1(K,T^\vee\otimes\mathbf{Q}_p/\mathbf{Z}_p)\rightarrow H^2(K,\mathbf{Q}_p/\mathbf{Z}_p(1))\cong\mathbf{Q}_p/\mathbf{Z}_p$$ the annihilator of $H^1_f(K,T)$ is $H^1_f(K,T^\vee\otimes\mathbf{Q}_p/\mathbf{Z}_p)$ (and vice versa) (where $T^\vee$ denotes the Tate dual). That the latter thing is the $H^1(K,E(\overline{K}))$ that you want it to be is equation (3.3) of the Bloch–Kato paper.

For example, when trying to generalize some construction using Kummer theory for elliptic curves to the case of higher weight modular forms, people use $H^1_f(K,T)$. Of course, the Bloch–Kato Selmer groups pretty much permeate a lot of things right now.